Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#559 ⋊ C2 |
| Sylow | C3 ⋊ #3510 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#559 | C2 |
| prime radical | #5715 | C2 |
| hypercenter | #559 | D3 |
| hypocenter | C3 | #3510 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #559 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(12,15,16,24)(13,19,18,23)(14,22,21,27)(17,25,20,26);(2,3)(14,21)(17,20)(22,27)(25,26);(4,7,8,6)(5,9,10,11)(13,18)(17,20)(19,23)(25,26);(2,3)(4,6,8,7)(5,9,10,11)(12,14,16,21)(13,17,18,20)(15,22,24,27)(19,25,23,26);(4,5)(6,9)(7,11)(8,10)(12,13)(14,20)(15,23)(16,18)(17,21)(19,24)(22,25)(26,27)Polycyclic presentation:
2^7,3|b^2=g,c^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df,h^b=h^2,h^d=h^2,h^e=h^269 conjugacy classes — view the full character table.
| maximal subgroup of | #11675233, #11675235, #11675237, … |
| maximal normal subgroup of | #11675233, #11675235, #11675237, … |
| maximal quotient of | #10955340, #10955594, #10955642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19628);